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  • Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 19

Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 19

Answers (1)

Answer: \frac{d y}{d x}=e^{x}+10^{x} \log 10+x^{x}(\log n+1)

Hint:  Differentiate the statement taking u and v

Given: y=e^{x}+10^{x}+x^{x}

Solution:  
Let x^{x}=u

        \begin{aligned} &y=e^{x}+10^{x}+u \\\\ &\frac{d y}{d x}=\frac{d\left(e^{x}\right)}{d x}+\frac{d\left(10^{x}\right)}{d x}+\frac{d u}{d x} \end{aligned}        ..........(1)

For \mathrm{u}, u=x^{x}

        \begin{aligned} &\log u=\log x^{x} \\\\ &\frac{d}{d x}(\log u)=x \log x \end{aligned}

        \frac{1}{u} \frac{d u}{d x}=\log x+x \cdot \frac{d}{d x}(\log x)

        \begin{aligned} &\frac{d u}{d x}=u\left[\log x+\not x \cdot \frac{1}{\not{x}}\right] \\\\ &=x^{x}(\log x+1) \end{aligned}

Put \frac{d u}{d x}  in eq (1)

        \begin{aligned} &\frac{d}{d x}\left(e^{x}\right)=e^{x} \\\\ &\frac{d y}{d x}=e^{x}+10^{x} \log 10+x^{x}(\log x+1) \end{aligned}

 

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